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Twarock, Reidun orcid.org/0000-0002-1824-2003, Valiunas, Motiejus and Zappa, Emilio (2015) Orbits of crystallographic embedding of non-crystallographic groups and applications to virology. Acta crystallographica. Section A, Foundations and advances. pp. 569-582. ISSN 2053-2733
https://doi.org/10.1107/S2053273315015326
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Acta Cryst. (2015). A71, 569–582 http://dx.doi.org/10.1107/S2053273315015326 569
Orbits of crystallographic embedding of non-crystallographic groups and applications to virology
Reidun Twarock,a,b,c Motiejus Valiunasd and Emilio Zappaa,c*
aDepartment of Mathematics, University of York, York, England, bDepartment of Biology, University of York, York,
England, cYork Centre for Complex Systems Analysis, University of York, York, England, and dFaculty of Mathematics,
University of Cambridge, Cambridge, England. *Correspondence e-mail: [email protected]
The architecture of infinite structures with non-crystallographic symmetries can
be modelled via aperiodic tilings, but a systematic construction method for finite
structures with non-crystallographic symmetry at different radial levels is still
lacking. This paper presents a group theoretical method for the construction of
finite nested point sets with non-crystallographic symmetry. Akin to the
construction of quasicrystals, a non-crystallographic group G is embedded into
the point group P of a higher-dimensional lattice and the chains of all G-
containing subgroups are constructed. The orbits of lattice points under such
subgroups are determined, and it is shown that their projection into a lower-
dimensionalG-invariant subspace consists of nested point sets withG-symmetry
at each radial level. The number of different radial levels is bounded by the
index of G in the subgroup of P. In the case of icosahedral symmetry, all
subgroup chains are determined explicitly and it is illustrated that these point
sets in projection provide blueprints that approximate the organization of
simple viral capsids, encoding information on the structural organization of
capsid proteins and the genomic material collectively, based on two case studies.
Contrary to the affine extensions previously introduced, these orbits endow
virus architecture with an underlying finite group structure, which lends itself
better to the modelling of dynamic properties than its infinite-dimensional
counterpart.
1. Introduction
Non-crystallographic symmetries are ubiquitous in physics,
chemistry and biology. Prominent examples are quasicrystals,
alloys exhibiting five-, eight-, ten- and 12-fold symmetry with
long-range order in their atomic organization (Steurer, 2004)
and, in carbon chemistry, icosahedral carbon cage structures
called fullerenes (Kroto et al., 1985), with architectures akin to
Buckminster Fuller’s geodesic domes. Icosahedral symmetry
also plays a fundamental role in virology. Viruses encapsulate
and hence protect their genomic material inside a protein
shell, called capsid, that in the vast majority of cases possesses
icosahedral symmetry. In 1962, Caspar and Klug proposed, in
their seminal paper (Caspar & Klug, 1962), a theory to
describe the geometry of icosahedral viral capsids and predict
the locations and orientations of the capsid proteins. Inspired
by the structure of the geodesic dome, they derived a series of
icosahedral triangulations, called deltahedra. More recently,
Twarock (2004) proposed a generalization of this theory by
considering more general tilings of the capsid surface inspired
by the theory of quasicrystals.
Caspar–Klug theory and generalizations thereof describe
the capsid of a virus as a two-dimensional object rather than in
three-dimensional space. Therefore, they do not provide
information about other important features of the capsid, such
ISSN 2053-2733
Received 19 May 2015
Accepted 17 August 2015
Edited by J.-G. Eon, Universidade Federal do Rio
de Janeiro, Brazil
Keywords: orbits; non-crystallographic
symmetry; icosahedral viruses; computational
group theory.
# 2015 International Union of Crystallography
as its thickness and the organization of the genomic material
encapsulated within. Experiments showed that many viruses
exhibit icosahedral symmetry at different radial levels:
examples are the dodecahedral cage of RNA observed in
Pariacoto Virus (Tang et al., 2001) and the double-shell
structure of the genomic material of Bacteriophage MS2
(Toropova et al., 2008). These results suggest that the
symmetry of the virus should be extended to include infor-
mation on the capsid proteins and the packaged genome
collectively.
A first step towards this goal was the principle of affine
extensions, described in a series of papers (Keef & Twarock,
2009; Keef et al., 2008, 2012). In this work, the generators of
the icosahedral group have been extended by a non-compact
generator acting as a translation, with the additional
requirement that the resulting words of the group satisfy non-
trivial relations. Such affine extension can also be obtained via
a construction similar to affine extensions in the theory of
Kac–Moody algebras (Carter, 2005). In this case, icosahedral
symmetry is extended via an extension of the Cartan matrix,
resulting in the addition of a non-compact operator to the
generators of the icosahedral group (Patera & Twarock, 2002;
Dechant et al., 2012, 2013). The orbits of the affine extensions
thus constructed consist of infinite sets of points that
densely fill the space, since the icosahedral group is non-
crystallographic in three dimensions. Since viral capsids are
finite objects, a cut-off parameter must be introduced, that
limits the number of monomials of the affine group. In
previous work, words characterized by a finite action of the
translation operator had been used to construct multi-shell
structures, in which each radial level displays icosahedral
symmetry. However, such a cut-off implies that the point sets
are not invariant under the extended group structure, which
limits the use of these concepts in the formulation of energy
functions, e.g. Hamiltonians or in the context of Ginzburg–
Landau theory.
An alternative to this approach based on affine extension is
Janner’s work, which models viral architecture in terms of
lattices. In a series of papers (Janner, 2006, 2010a,b, 2011a,b),
Janner embedded virus structure into lattices, and showed that
this provided an approximation for virus architecture, and
provides an alternative approach for the modelling of the
onion-like fullerenes (Janner, 2014); a paper combining this
lattice approach with the affine extensions mentioned above
was also published (Janner, 2013). Subsequently, approxima-
tions of virus architecture in terms of quasilattices were
developed (Salthouse et al., 2015), which provide an alter-
native to the lattice approach of Janner, and by construction
have vertex sets that contain the point arrays determined by
the affine extended groups as subsets. All of these approaches
approximate viruses in terms of infinite structures, lattices,
quasilattices, infinite groups, which require a cut-off. In the
case of the lattices and quasilattices, the cut-off consists of
choosing a subset of the infinite (quasi)lattice, and in the case
of affine extensions the action of the translation operator has
to be limited. This is the motivation for the study described in
the present paper in which we develop an approach in terms
of mathematical concepts that are intrinsically finite-dimen-
sional, because they are related to orbits of finite groups.
In this paper we introduce a new group theoretical method
to study nested point sets with non-crystallographic symme-
tries, based on the embedding of non-crystallographic groups
into higher-dimensional lattices (Senechal, 1995). This
embedding is a standard way to define mathematically
quasicrystals, e.g. via the cut-and-project schemes and model
sets (Moody, 2000), or the superspace approach (Janssen &
Janner, 2014). More generally, in order to model objects in
three dimensions that possess a non-crystallographic
symmetry at different radial levels, it makes sense to embed
the non-crystallographic symmetry into a crystallographic
setting and use the long-range order implied to induce in
projection information on the collective arrangements of
different radial levels. Janner (2008) gave a first approach in
this direction, by analysing double-shell structures with five-
fold symmetry as projected orbits of specific point groups in
higher dimensions. Here we present a more systematic study
for general non-crystallographic symmetries. Specifically, we
embed a non-crystallographic group G into the point group Pof a lattice in the minimal higher dimension where the cut-
and-project construction is possible. Since this embedding is
not, in general, maximal, we consider the subgroups K of Pcontaining G as a subgroup, which extend the symmetry
described by G. We prove that the projection of the orbits of
lattice points under such subgroups into a lower-dimensional
subspace invariant under G is a nested finite point set with
non-crystallographic symmetryG. We show that the number of
distinct radial levels in the projected orbits is bounded by the
index of G in K.
As an illustration of this approach, we provide analytically
an explicit construction of planar nested structures for non-
crystallographic dihedral groups. Moreover, in order to pave
the way for applications to icosahedral viruses, we apply this
approach to the icosahedral group I , which can be embedded
into the point group of the simple cubic lattice in six dimen-
sions (Zappa et al., 2014). We classify all the I -containingsubgroups of the hyperoctahedral group, with the aid of the
software GAP (The Gap Group, 2013), which is designed for
problems in computational group theory. Since the six-
dimensional lattice is infinite, a cut-off parameter must be
introduced in order to select a finite number of lattice points
whose orbits can then be used to model the capsid. By
construction, all point arrays have full icosahedral symmetry,
i.e. containing reflections as well as rotations. Since viruses are
known to be chiral, this may seem perplexing; however, we
note that point arrays do not fully constrain viral architecture,
and thus proteins can be positioned in the capsid so as to
break the full symmetry, as long as they adhere overall to the
blueprint indicated by the points. Therefore, it is not possible
to obtain a full classification of the orbits as was done by Keef
et al. (2012). However, these results provide for the first time a
finite group structure, albeit in a higher-dimensional space,
underlying the geometry of the multiple layers of material in a
virus. This has important consequences for the modelling of
physical properties; specifically, conformational changes of
570 Reidun Twarock et al. ! Orbits of non-crystallographic groups and viruses Acta Cryst. (2015). A71, 569–582
research papers
viral capsids, which are important for the virus to become
infective, can be modelled in the framework of the Ginzburg–
Landau theory of phase transitions (Zappa et al., 2013), via the
formulation of an energy function invariant under the
generators of the symmetry group of the capsid.
The paper is organized as follows. After reviewing, in x2, theembedding of non-crystallographic groups into higher-
dimensional lattices, in x3 we describe the new group theo-
retical setup to model finite nested structures with non-
crystallographic symmetry. As a first application, we study in
x4 planar nested point sets obtained from projection of
extensions of embedded non-crystallographic dihedral groups.
In x5 we analyse in detail the case of icosahedral symmetry,
classifying the chain of subgroups containing the icosahedral
group embedded into the six-dimensional hyperoctahedral
group. Finally, in x6 we use these results to obtain geometric
constraints on viral capsid architecture, and present two case
studies, namely the capsids of Pariacoto Virus and Bacter-
iophage MS2, whose structures have been intensively studied
experimentally. We conclude in x7 by discussing further
applications of these results.
2. Crystallographic embedding of non-crystallographicgroups
Our new group theoretical setup relies on the embedding of
non-crystallographic symmetries into the point group of
higher-dimensional lattices. This is a standard method in the
theory of quasicrystals; here we briefly review it and fix the
notation that we are going to use throughout the paper. We
refer to Senechal (1995) and Baake & Grimm (2013) for
further information.
The point group P of a lattice L in Rd with generator matrix
B is the maximal set of orthogonal transformations that leave
the lattice invariant:
PðLÞ :¼ fQ 2 OðdÞ : 9M 2 GLðd;RÞ : QB ¼ BMg: ð1Þ
P is a finite group and does not depend on the matrix B
(Senechal, 1995). The lattice group ! constitutes an integral
representation of the point group P with respect to B:
!ðBÞ :¼ fM 2 GLðd;ZÞ : 9Q 2 PðLÞ : B%1QB ¼ Mg: ð2Þ
A finite group of isometries G is non-crystallographic in
dimension k if it does not leave any lattice invariant in Rk.
Following Levitov & Rhyner (1988), we introduce the
following:
Definition 2.1. Let G & OðkÞ be a finite non-crystal-
lographic group of isometries. A crystallographic representa-
tion of G is a matrix group eGG satisfying the following
conditions:
(C1) eGG stabilizes a lattice L in Rd, with d> k, i.e. eGG is a
subgroup of the point group P of L;(C2) eGG is reducible in GLðd;RÞ and contains an irreducible
representation (irrep) !k of G of degree k, i.e.
eGG ’ !k ' !0; degð!0Þ ¼ d% k: ð3Þ
The condition (C1) implies that the matrices representing
the elements of eGG with respect to a generator matrix B of the
lattice are integral or, equivalently, B%1eGGB is a subgroup of
the lattice group ! & GLðd;ZÞ of L [cf. equation (2)]. As a
consequence, the character "eGG
is an integer-valued function.
The condition (C2) is necessary for the construction of
quasicrystals in Rkvia the cut-and-project method (Senechal,
1995; Baake & Grimm, 2013).
The minimal dimension d> k for which a crystallographic
representation eGG of G is possible is called the minimal crys-
tallographic dimension of G. The conditions "eGG2 Z and
equation (3) can be easily verified with the aid of the character
table of G and Maschke’s theorem (Fulton & Harris, 1991).
The existence, and possibly an explicit construction, of lattices
in Rd whose point group contains a crystallographic repre-
sentation of G is a more difficult task. In the case of icosa-
hedral symmetry, the minimal crystallographic dimension is six
and the lattices in R6 have been classified in Levitov & Rhyner
(1988) (this is explained in more detail in x5). For planar non-crystallographic symmetries described by the dihedral groups
D2n, the minimal crystallographic dimension is ’ðnÞ, the Euler
function of n. We will go back to this example in x4.Let us denote by VðkÞ the invariant subspace of Rd which
carries the irrep !k of G. Let # : Rd ! VðkÞ be the projection
into V ðkÞ, i.e. the linear operator such that the diagram
Rd eGGðgÞ%! Rd
## ##
VðkÞ !kðgÞ%! VðkÞ
ð4Þ
commutes for all g 2 G:
#ðeGGðgÞvÞ ¼ !kðgÞð#ðvÞÞ; 8g 2 G; 8v 2 Rd: ð5Þ
Let V ðd%kÞ denote the orthogonal complement of V ðkÞ in Rd.
We recall the following proposition [for the proof, see Sene-
chal (1995)]:
Proposition 2.1. The following are equivalent:
(i) Vðd%kÞ is totally irrational, i.e. Vðd%kÞ \ L ¼ f0g;(ii) # jL is one-to-one.
The triple ½VðkÞ;Vðd%kÞ;L) is the starting point to define
model sets via cut-and-project schemes with G-symmetry
(Moody, 2000), which is a standard way to define quasicrystals
mathematically. In this paper, we construct finite point sets
resulting from the projection into VðkÞ of orbits of points of Lunder G-containing subgroups of the point group P. We
explain this construction in the next section.
3. Nested point sets obtained from projection
The embedding of a non-crystallographic group G into a
higher-dimensional lattice L is, in general, not maximal. This
research papers
Acta Cryst. (2015). A71, 569–582 Reidun Twarock et al. ! Orbits of non-crystallographic groups and viruses 571
means that there exist proper subgroups of the point group Pof L which contain a crystallographic representation eGG ofG as
a subgroup. Therefore, we introduce the set
AeGG:¼ fK * P : eGG * Kg; ð6Þ
which consists of all the eGG-containing subgroups of P. Forcomputational purposes, we fix the generator matrix B of L,and consider the subgroup structure of the lattice group ! in
that representation, i.e. the set AHðBÞ ¼ fK<! : H<Kg,with H :¼ B%1
eGGB. Notice that a different choice in the
generator matrix of the lattice results in subgroups K0 conju-
gate to K in GLðd;ZÞ.
The elements in AH encode the symmetry described by G
plus additional generators that extend this symmetry. LetK be
an element of AH and let n :¼ ½K : H) be the index ofH in K.
Let T ¼ fg1; . . . ; gng be a transversal ofH inK, i.e. a system of
representatives in K of the right cosets of H in K (Holt et al.,
2005). Let v 2 L be a lattice point, which can be written as
v ¼ ðm1; . . . ;mdÞ, with mi 2 Z (since we fixed the basis B). vcan be taken as seed point for the orbit OKðvÞ ¼ fkv : k 2 Kgunder K. With this setup, we prove the following theorem.
Theorem 3.1. Let OiðvÞ + OHgiðvÞ ¼ fhgiv : h 2 Hg be the
orbit of v 2 L with respect to the coset Hgi, and let us denote
by PiðvÞ :¼ #ðOiðvÞÞ the orbit projected into VðkÞ, the subspace
of dimension k carrying the irrep !k of G [cf. equation (4)].
Then we have:
(i) PiðvÞ is well defined, i.e. does not depend on the choice of
the transversal T.
(ii) PiðvÞ retains the symmetry described by G.
(iii) PiðvÞ ¼ PjðvÞ if and only if
g%1j Hgi \ StabKðvÞ 6¼ ;: ð7Þ
(iv) If H is normal in K, then all PiðvÞ have the same
cardinality.
Proof. (i) Let T 0 ¼ ðg01; . . . ; g0nÞ be another transversal forH
inK. This implies that there exist hhi 2 H, for i ¼ 1; . . . ; n, such
that g0i ¼ hhigi. We have
O0iðvÞ ¼ OHg0
iðvÞ ¼ fhg0i : h 2 Hg ¼ fhhhigiv : h 2 Hg
¼ OiðvÞ; i ¼ 1; . . . ; n;
and the result follows.
(ii) It follows from the commutative property in (5); in
particular, we have
#ðOiðvÞÞ ¼ f#ðhgivÞ : h 2 Hg ¼ f!kðhÞ#ðgivÞ : h 2 Hg¼ fhh#ðgivÞ : hh 2 !kg ¼: OOiðvÞ;
for i ¼ 1; . . . ; n. The orbit OOiðvÞ has G-symmetry by
construction.
(iii) We have
PiðvÞ ¼ PjðvÞ()#ðOiðvÞÞ ¼ #ðOjðvÞÞ()OiðvÞ
¼ðsince# is 1%1Þ OjðvÞ()fhgiv; h 2 Hg ¼ fhgjv : h 2 Hg()9h; k 2 H : hgiv ¼ kgjv()g%1
j k%1hgiv ¼ v
()g%1j k%1hgi 2 StabKðvÞ;
which proves the statement.
(iv) SinceH is normal inK, the cosetsHgi form the quotient
groupK=H of size n. LetX :¼ fOi : i ¼ 1; . . . ; ng be the set ofall the orbits with respect to the cosets Hgi. In the following,
we will omit the dependence on v for ease of notation. We can
define an action of K=H on X as Hi ,OHj:¼ OHiHj
. This
action is well defined since K=H is a group, and it is transitive
since, for every element OHi2 X , we haveHj ,OH%1
j Hi¼ OHi
.
Let SH :¼ StabK=HðOHÞ denote the stabilizer ofOH under this
action. With s :¼ jSHj we thus have by the orbit–stabilizer
theorem
r :¼ jXj ¼ jK=HjjSHj
¼n
s:
It follows that the setsOiðvÞ are in bijection with the left cosets
of SH in K=H. We denote these cosets by Ai, for i ¼ 1; . . . ; r.
These form a partition of the quotient group K=H, which we
write as
Hð1Þ1 ; . . . ;Hð1Þ
s|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
A1
; . . . ;HðiÞ1 ; . . . ;HðiÞ
s|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
Ai
; . . . ;HðrÞ1 ; . . . ;HðrÞ
s|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
Ar
:
By construction,OHðiÞ
j
¼ OHðiÞ
k
, for j; k ¼ 1; . . . ; s. Let us define
the sets
KðiÞ :¼[
s
j¼1
HðiÞj & K; i ¼ 1; . . . ; r: ð8Þ
The set fKðiÞ : i ¼ 1; . . . ; rg constitutes a partition of K, since itis the union of cosets. Moreover they all have the same order:
jKðiÞj ¼ s , jHj ¼: N; 8i ¼ 1; . . . ; r: ð9Þ
Let S ¼ ðHð1Þ1 ; . . . ;HðrÞ
1 Þ be a transversal for the coset of SH in
K=H. It follows from equation (8) that KðiÞ ¼
fk 2 K : kv 2 OHðiÞ
1
g; therefore
OKðiÞ ¼ fkv : k 2 KðiÞg ¼ OHðiÞ
1
:
To conclude, we observe that each KðiÞ contains complete
cosets of K=StabKðvÞ. In fact, let k StabKðvÞ be a coset in
K=StabKðvÞ. If k 2 KðiÞ, then an element in k StabKðvÞ is of
the form kkk, with kk 2 StabKðvÞ, and belongs to KðiÞ since
ðkkkÞv ¼ kðkkvÞ ¼ kv 2 OHðiÞ
1
. Therefore, eachKðiÞ is partitioned
into jKðiÞj=jStabKðvÞj sets: each of these sets corresponds to
a distinct point in the orbit OHðiÞ
1
. Since jKðiÞj ¼ N for all i
due to equation (9), each orbit OHðiÞ
1
has the same number
of points, and hence also each PiðvÞ, because the projection
is one-to-one. &
The decomposition of K 2 AH into cosets with respect toHinduces a well defined decomposition of the projected orbit
#ðOKðvÞÞ [cf. equation (4)]:
572 Reidun Twarock et al. ! Orbits of non-crystallographic groups and viruses Acta Cryst. (2015). A71, 569–582
research papers
#ðOKðvÞÞ ¼[
n
i¼1
#ðOiðvÞÞ ¼[
n
i¼1
O!kð#ðgivÞÞ: ð10Þ
The point set defined by equation (10) consists of points
situated at different radial levels, since, in general,
j#ðgivÞj 6¼ j#ðgjvÞj for i 6¼ j, where j , j denotes the standard
Euclidean norm in Rk. Hence the projected orbit is an onion-
like structure, with each layer being the union of the projec-
tion of orbits corresponding to different cosets. It follows that
the number r of distinct radial levels is bounded by the index
of H in K.
Using these results, we can set up a procedure to extend the
non-crystallographic symmetry described by !k in VðkÞ. In
particular, let x 2 VðkÞ be a seed point for the orbit of !k. The
pre-image v ¼ #%1ðxÞ is a point of the lattice L by construc-
tion. Let K be an element of AH. The projection of OKðvÞ
contains the orbit O!kðxÞ, which corresponds to the coset H
[compare with equation (10)], and possibly more layers with
G-symmetry. This procedure can be iterated; let us consider
the chain of subgroups in AH:
H & K1 & K2 & . . . & Km & !:
By ascending the chain we obtain a chain of orbits
OKiðvÞ & OKiþ1
ðvÞ; the projection of such orbits into VðkÞ
induces a chain of nested shells. We can summarize the
situation in the following diagram:
OHðvÞ & L %!extend OK1ðvÞ %! . . .%! O
!ðvÞ
"lift #project # #
O!kðxÞ %! #ðOK1
ðvÞÞ . O!kðxÞ %! . . .%! #ðO
!ðvÞÞ . . . . . O!k
ðxÞ
:
ð11Þ
In the next section we present a first application of these
results in the case of planar non-crystallographic symmetries.
4. Embedding of dihedral groups D2n and planar nestedstructures
Let n> 0 be a natural number. The dihedral group D2n is the
symmetry group of a regular n-gon, and consists of n rotations
and n reflections, with presentation (Holt et al., 2005)
D2n ¼ hRn; S : Rnn ¼ e; SRn ¼ R%1
n Si; ð12Þ
where Rn is a rotation by 2#=n, and S a reflection.
Let $n ¼ exp½ð2#iÞ=n) 2 C be a primitive root of unity, and
let Z½$n) be the ring of integers of the fieldQð$nÞ. The standard
embedding of D2n into a ’ðnÞ-dimensional lattice, where ’ðnÞ
denotes the Euler function, is achieved via the Minkowski
embedding of Z½$n) (Baake & Grimm, 2013). Specifically, let Gdenote the Galois group of Qð$nÞ. G is isomorphic to
Z/n :¼ fm 2 Zn : gcd ðm; nÞ ¼ 1g, the multiplicative group of
Zn, and therefore consists of ’ðnÞ elements. Such elements are
automorphisms of Qð$nÞ given by $n 7! $mn , where n and m are
coprime, and they are pairwise conjugate. We can then choose
’ðnÞ=2 non-conjugate elements %i in G, where %1 is the identity.The Minkowski embedding of Z½$n) is then given by
L’ðnÞ :¼ fðx; %2ðxÞ; . . . ; %’ðnÞ2Þ : x 2 Z½$n)g & C
’ðnÞ2 ’ R’ðnÞ;
ð13Þ
which is a lattice in R’ðnÞ. The projection # : L’ðnÞ ! C on the
first coordinate is, by construction, one-to-one on its image
#ðL’ðnÞÞ ¼ Z½$n).
We can define an action ofD2n in Z½$n) in the following way:
Rn , x ¼ $nx; S , x ¼ x;
where x denotes the complex conjugation in C and x 2 Z½$n).
Note that this action is well defined as every element of D2n
stabilizes Z½$n). If g is an element of I2ðnÞ, g can be lifted to an
element ~gg defined by
~gg , #%1ðxÞ ¼ #%1ðg , xÞ; ð14Þ
which is well defined since the projection is one-to-one. In
particular, we have
~RRn , #%1ðxÞ ¼ #%1ðRn , xÞ ¼ #%1ð$nxÞ
¼ ð$nx; %2ð$nxÞ; . . . ; %’ðnÞ2ð$nxÞÞ:
Similarly we have
~SS , #%1ðxÞ ¼ #%1ðS , xÞ ¼ #%1ðxÞ ¼ x; %2ðxÞ; . . . ; %’ðnÞ2ðxÞ
" #
:
It follows that the transformations ~RRn and ~SS are orthogonal
and stabilize the lattice L’ðnÞ. Therefore the set f~gg : g 2 D2ng isan embedding of D2n into the point group of L’ðnÞ. We point
out that, although this construction is a priori possible and well
defined for all natural numbers, it is difficult to find the explicit
form of the point group of L’ðnÞ in equation (13) for general n.
The explicit form is known, in particular, for n ¼ 5, 8 and 12
(Baake & Grimm, 2013).
We now prove the existence of an extension K of D2n
embedded into PðL’ðnÞÞ, i.e. a subgroup K of PðL’ðnÞÞ such that
D2n is a normal subgroup of K. Note that D2n can be seen as a
subgroup of the symmetric group Sn, acting on the vertices of
a regular n-gon. More precisely, let R0n ¼ ð1; 2; . . . ; nÞ be an
n-cycle and let S0 be the permutation defined by
S0ðjÞ ¼ %jmod n, for j ¼ 1; . . . ; n; then hR0n; S
0i defines a
permutation representation of D2n. Let T ¼ hR0ni ’ Zn, and
define K as the normalizer (Artin, 1991) of T:
K :¼ NSnðTÞ ¼ f% 2 Sn : %
%1T% ¼ Tg: ð15Þ
We point out that K thus constructed is referred to as the
holomorph of the group Zn, and denoted by HolðZnÞ (Hall,
1959). We have the following:
Lemma 4.1. D2n is a proper normal subgroup of
K ¼ HolðZnÞ when n ¼ 5 or n 0 7.
Proof. We have, using notation as in equation (15),
% 2 K () %T%%1 ¼ T () %Rn%%1 2 T ()
%ð1; 2; . . . ; nÞ%%1 ¼ ð1; 2; . . . ; nÞm
for some m 2 Zn ()ð%ð1Þ; %ð2Þ; . . . ; %ðnÞÞ ¼ ð1; 2; . . . ; nÞ
mfor some m 2 Zn with
gcd ðm; nÞ ¼ 1 [otherwise ð1; 2; . . . ; nÞm
decomposes into
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disjoint cycles] () 8j 2 Zn; %ðjÞ ¼ mjþ l for some m; l 2 Zn
with gcd ðm; nÞ ¼ 1. To sum up:
K ¼ f% 2 Sn : 9m 2 Z/n ; l 2 Zn : %ðjÞ ¼ mjþ l; 8j 2 Zng:
K contains R0n and S0, which correspond to m ¼ 1, l ¼ 1 and
m ¼ %1, l ¼ 0, respectively. It follows that D2n is a subgroup
of K. We notice that jKj ¼ ’ðnÞn, which is greater than 2n for
n ¼ 5 or n 0 7. Hence D2n is a proper subgroup of K for these
values of n, which correspond to the non-crystallographic
cases.
In order to prove the normality, we write
D2n ’ hR0ni [ hR0
niS0 ¼ T [ TS0:
Define % 2 K by %ðjÞ ¼ mjþ l. We want to prove that
%D2n ¼ D2n%. Clearly %T ¼ T% by definition of K [cf. equa-
tion (15)]. We are then left to show that %TS0 ¼ TS0%. For
s; j 2 Zn we have ððR0nÞ
sS0ÞðjÞ ¼ s% j, which implies
ð%ððR0nÞ
sS0Þ%%1ÞðjÞ ¼ ms% jþ 2l. Therefore, %ððR0
nÞsS0Þ%%1 ¼
ðR0nÞ
msþ2lS0, hence %TS0 ¼ TS0%, and the result follows. &
We now prove the following:
Proposition 4.1. K ¼ HolðZnÞ is a subgroup of PðL’ðnÞÞ.
Proof. Let us define the functions tm;l 2 AutðZ½$n)Þ by
tm;l
X
n%1
j¼0
aj$jn
!
:¼X
n%1
j¼0
aj$mjþln ;m 2 Z/
n ; l 2 Zn: ð16Þ
Notice that the elements tm;0, with m 2 Z/n , correspond to the
Galois automorphisms %m, which constitute the Galois group Gof Qð$nÞ. Let K
0 ¼ ftm;l : m 2 Z/n ; l 2 Zng. K0 is a G-containing
subgroup of AutðZ½$n)Þ. In particular, composition of two
elements is given by
tm;l , tm0;l0 ¼ tmm0;ml0þl; ð17Þ
and the inverse of an element tm;l is tm%1;%m%1l. Let & : K ! K0
be the function
&ð%Þ ¼ tm;l; %ðjÞ ¼ mjþ l:
& is an isomorphism by construction. We defineD02n :¼ &ðD2nÞ.
By Lemma 4.1, we have that D02n is a normal subgroup of
K0 <AutðZ½$n)Þ.
We can write the Minkowski embedding of Z½$n) as L’ðnÞ ¼
fty1;0ðzÞ; . . . ; ty’ðnÞ=2;0ðzÞ : z 2 Z½$n)g 2 C12’ðnÞ ffi R’ðnÞ, where 1 =
y1 < , , , < y’ðnÞ=2 < n=2 and gcdðyj; nÞ ¼ 1, for all j. We can
then lift tm;l as in equation (14), and obtain
~ttm;l , #%1ðzÞ ¼ #%1ðtm;lðzÞÞ ¼ tm;0ðtm;lðzÞÞ; . . . ; tmy’ðnÞ=2;0
ðtm;lðzÞÞ" #
¼½by ð17Þ)
tmy1;y1lðzÞ; . . . ; tmy’ðnÞ=2;y’ðnÞ=2
ðzÞ" #
¼½by ð16Þ)
$y1ln tmy1;0ðzÞ; . . . ; $
y’ðnÞ=2ln tmy’ðnÞ=2
ðzÞ" #
:
Therefore ~ttm;l just rearranges the coordinates of #%1ðzÞ,
possibly converting some of them to their complex conjugates
and/or multiplying them by a power of $n. Hence K0 stabilizes
the lattice Ln and this action is orthogonal. Thus K0 is a
subgroup of PðLnÞ and the result is proved. &
It follows that we can construct nested point sets with n-fold
symmetry using the extension K ¼ HolðZnÞ and the
Minkowski embedding L’ðnÞ. The number r of distinct radial
levels obtained via projection is at most
r * ½K : H) ¼’ðnÞn
2n¼
’ðnÞ
2:
4.1. Fivefold symmetry
As a first example, we consider the case n ¼ 5. In this case,
the Minkowski embedding of D10 is isomorphic to the root
lattice A4 (Baake & Grimm, 2013), whose simple roots are
given by 'i ¼ ei % eiþ1, for i ¼ 1; . . . ; 4, and ei denotes the
standard basis of R5 [cf. also Carter (2005) for more details on
root systems of semisimple Lie algebras]. With respect to the
basis of simple roots, we obtain a representation H of D10
which is a subgroup of the lattice group !ðA4Þ (which is
isomorphic to the symmetric group S5):
H ¼
1 0 0 0
1 0 0 %1
1 0 %1 0
1 %1 0 0
0
B
B
@
1
C
C
A
;
%1 1 0 0
0 1 0 0
0 1 0 %1
0 1 %1 0
0
B
B
@
1
C
C
A
* +
: ð18Þ
This representation splits into two two-dimensional irreps,
which induce a decomposition R4 ’ Eð1Þ ' Eð2Þ, where Eð1Þ
and Eð2Þ are both totally irrational with respect to the root
lattice A4. A basis for each of them can be found using tools
from the representation theory of finite groups (Fulton &
Harris, 1991). The projection #ð1Þ2 : R
4%!Eð1Þ is given by
#ð1Þ2 ¼
1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð3% (Þp
%(0ffiffiffiffiffiffiffiffiffiffiffi
3% (p ffiffiffiffiffiffiffiffiffiffiffi
3% (p
0 %ffiffiffiffiffiffiffiffiffiffiffi
3% (p
%1 2% ( %2(0 2% (
% &
;
ð19Þ
where ( :¼ 12 ð1þ
ffiffiffi
5p
Þ denotes the golden ratio and
(0 :¼ 1% ( its Galois conjugate. The space Eð1Þ carries the
irrep !2:
!2 ¼1 0
0 %1
% &
;1
2
%(0ffiffiffiffiffiffiffiffiffiffiffi
( þ 2p
ffiffiffiffiffiffiffiffiffiffiffi
( þ 2p
(0
% &' (
: ð20Þ
With the help of GAP, we study the set AH of subgroups of
!ðA4Þ containing H [compare with equation (6)]. There is a
unique chain of subgroups containing a proper extension ofH:
H / K & !ðA4Þ; ð21Þ
where K is, in fact, isomorphic to HolðZ5Þ. The explicit
representation of K is given by
574 Reidun Twarock et al. ! Orbits of non-crystallographic groups and viruses Acta Cryst. (2015). A71, 569–582
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K ¼
*
0 %1 1 0
%1 0 1 0
0 0 1 0
0 0 1 %1
0
B
B
B
@
1
C
C
C
A
;
0 0 0 %1
1 0 0 %1
0 1 0 %1
0 0 1 %1
0
B
B
B
@
1
C
C
C
A
;
1 0 0 0
1 0 %1 1
0 1 %1 1
0 1 %1 0
0
B
B
B
@
1
C
C
C
A
+
:
The groupK corresponds to the point group 54 given in Janner
(2010b). We point out that, by Theorem 3.1, the point sets
obtained from projection of orbits of points of the root lattice
A4 consist of at most two radial levels, which can either be two
nested decagons or two nested pentagons. In Fig. 1 we show an
example of such point sets.
5. Nested point sets with icosahedral symmetry
As mentioned in x1, icosahedral symmetry plays a funda-
mental role in virology, carbon chemistry and quasicrystals.
For applications in the natural sciences, it is important to
distinguish between chiral and achiral symmetry. Chiral
icosahedral symmetry is described by the icosahedral group I ,which consists of all the rotations that leave an icosahedron
invariant and admits the presentation
I ¼ hg2; g3 : g22 ¼ g33 ¼ ðg2g3Þ5¼ ei;
where g2 and g3 are a two- and threefold rotation, respectively.
It has order 60 and it is isomorphic to the alternating group A5.
On the other hand, achiral icosahedral symmetry corresponds
to the full symmetries of an icosahedron (i.e. reflections
included), and it is described by the Coxeter group H3, whose
order is 120 and it is isomorphic to I / Z2.
For applications in virology, we focus firstly on chiral
icosahedral symmetry, since not all viral capsids are invariant
under reflections. Since the icosahedral group contains five-
fold symmetry, it is not crystallographic in three dimensions.
Its minimal crystallographic dimension, in the sense of Defi-
nition 2.1, is six (Senechal, 1995). In particular, there are
exactly three Bravais lattices left invar-
iant by I in R6, namely the simple cubic
(SC), face-centred cubic (f.c.c.) and
body-centred cubic (b.c.c.) lattices
(Levitov & Rhyner, 1988). The point
group of these lattices is the hyper-
octahedral group in six dimensions,
which we denote by B6 [cf. equation
(1)],
B6 ¼ fQ 2 Oð6Þ : Q ¼ M 2 GLð6;ZÞg¼ Oð6Þ \GLð6;ZÞ;
which consists of all the 6/ 6 ortho-
gonal and integral matrices. It is
isomorphic to the wreath product
Z2 o S6, where S6 denotes the symmetric
group on six elements (see Appendix A). Its order is 266! ¼
46 080. In what follows, we will focus on the SC lattice:
LSC :¼M
6
i¼1
Zei;
where ei, i ¼ 1; . . . ; 6, is the standard basis of R6; its point and
lattice groups coincide [cf. equation (1) and equation (2)]. The
crystallographic representations of I have been classified in
Zappa et al. (2014). They are all conjugated in B6; a repre-
sentative II can be chosen as the following:
II ðg2Þ ¼
0 0 0 0 0 1
0 0 0 0 1 0
0 0 %1 0 0 0
0 0 0 %1 0 0
0 1 0 0 0 0
1 0 0 0 0 0
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
;
II ðg3Þ ¼
0 0 0 0 0 1
0 0 0 1 0 0
0 %1 0 0 0 0
0 0 %1 0 0 0
1 0 0 0 0 0
0 0 0 0 1 0
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
: ð22Þ
II leaves two three-dimensional subspaces invariant, denoted
by Ek and E?, that are both totally irrational with respect to
the SC lattice. It follows that II decomposes, in GLð6;RÞ, into
two three-dimensional irreps, usually denoted by T1 and T2.
An explicit form of T1, useful for computations, is given by
T1ðg2Þ ¼1
2
%(0 1 (
1 %( %(0
( %(0 %1
0
B
@
1
C
A;
T1ðg3Þ ¼1
2
( %(0 1
(0 %1 (
1 %( (0
0
B
@
1
C
A: ð23Þ
The projection #k : R6 ! Ek is given by
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Figure 1Examples of point sets with fivefold symmetry via projection of orbits of points from the A4 rootlattice in R4. The lattice point v ¼ ð1; 2; 4; 3Þ (whose coordinates are written with respect to thebasis of simple roots) is taken as seed point from the orbits under the groups [cf. equation (21)]: (a)H ’ D10, (b) K ’ HolðZ5Þ and (c) !ðA4Þ ’ S5.
#k ¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð2þ (Þp
( 0 %1 0 ( 1
1 ( 0 %( %1 0
0 1 ( 1 0 (
0
@
1
A: ð24Þ
For achiral icosahedral symmetry, the crystallographic
representations of H3 are easily computed using the direct
product structure I / Z2. Specifically, if " ¼ f1;%1g is the
non-trivial irrep of Z2, then the representation II 2 " is a
crystallographic representation of H3, and is such that
II 2 " ’ T1 2 "' T2 2 " in GLð6;RÞ. We point out that
there exist other crystallographic groups in six dimensions
which contain the icosahedral group as a subgroup, and these
can be found using the GAP package CARAT (Opgenorth et
al., 1998). However, the representations of I induced by this
embedding do not split into two three-dimensional irreps of I,according to the classification provided by Levitov & Rhyner
(1988), and hence they are not suitable for the construction of
nested point sets by projection presented here.
In order to construct nested structures with icosahedral
symmetry, we consider the set of all the II -containingsubgroups of B6 [cf. equation (6)]:
AII :¼ fK<B6 : II <Kg: ð25Þ
With the help of GAP, it is possible to compute the set AII . In
order to make computations efficient, we use some results
from group theory. In particular, we recall that, if G is a
soluble group, then every subgroup of G is soluble
(Humphreys, 1996). Since the icosahedral group is isomorphic
to A5, it is not soluble. Therefore, any subgroup H of B6
containing II as a subgroup must not be soluble. Moreover, it
cannot be Abelian (since I is not) and the order of H must be
divisible by jI j ¼ 60, as a consequence of Lagrange’s theorem.
With these considerations, we provide the following algorithm.
Algorithm 5.1. In order to determine AII , perform the
following steps:
1. Compute the conjugacy classes Ci of the subgroups of B6.
2. List a representative Ki for each class Ci.
3. Rule out those representatives which have one of the
following properties:
(a) Ki is soluble;
(b) Ki is Abelian;
(c) 60 6 j jKij.4. For each Ki not ruled out, compute all the elements
Gi 2 Ci. If II <Gi, then add Gi to AII .
The algorithm was implemented in GAP and the results are
given in Table 1. There are 13 elements in AII , which we
denote by Gi, for i ¼ 1; . . . ; 13. A set of generators for each
group Gi is given in Appendix A. Clearly, G1 and G13 are the
icosahedral and hyperoctahedral group, respectively, whereas
G2 is isomorphic to H3. In Fig. 2 we show the graph of
inclusions of the groups Gi.
The projections into Ek of the orbits of lattice points under
the groups Gi produce nested point sets with icosahedral
symmetry at each radial level. An example is given in Fig. 3.
Every radial level corresponds to the union of cosets of Gi
with respect to II . It is worth pointing out that every group Gi,
for i> 3, contains H3 as well as I as subgroups. From a
geometrical point of view, this implies that the resulting orbits
in projection are all invariant under reflections, i.e. each radial
level possesses full icosahedral symmetry H3. This observation
provides a sharper bound on the number of distinct radial
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Table 1Classification of the subgroups of B6 containing the icosahedral group Ias a subgroup.
Subgroup Order Index
G1 ’ I 60 1G2 ’ H3 120 2G3 240 4G4 1920 32G5 3840 64G6 3840 64G7 3840 64G8 7680 128G9 11520 192G10 23040 384G11 23040 384G12 23040 384G13 ¼ B6 46080 768
Figure 2Graph of inclusions of the subgroups containing the icosahedral groupembedded into the hyperoctahedral group.
levels in projection: in fact, this is given by n=2, which is the
index of H3 in Gi (recall that n is the index of I in Gi).
6. Applications to viral capsid architecture
In this section we show that this group theoretical setup is a
powerful tool to rationalize viral architecture. Specifically, the
classification of the chains of subgroups of B6 extending
icosahedral symmetry, derived in x5, provides a suitable
mathematical framework to understand structural constraints
on viral capsids. As a first step towards this goal, we identify a
finite library of point arrays, corresponding to the projected
orbits of six-dimensional lattice points under the groups Gi
previously classified. Elements in this library depend on two
quantities: the group Gi 2 AII and the lattice point v 2 LSC.
The Gi are provided by our classification. As can be seen from
Fig. 2 and Table 1, the first group that gives icosahedral nested
shells in projection is G3. The index of G3 with respect to H3 is
2; therefore the number of radial levels is at most two. In order
to obtain deeper information about the geometry of capsids,
more radial levels are necessary. Therefore, we neglect the
orbits of G3 and consider the subgroups Gi, for i ¼ 4; . . . ; 13.
Moreover, v is chosen as follows: since the six-dimensional
lattice is infinite, we introduce a cut-off parameter N> 0 and
consider all lattice points within a six-dimensional cube,
I6N :¼ ½%N;N) / . . ./ ½%N;N) ¼ ½%N;N)6 & LSC;
containing ð2N þ 1Þ6lattice points. In particular, we consider
all orbits of the groups Gi within a bounded area around the
origin defined by N.
Based on this setup, the library of point arrays is obtained
via the action of the group Gi on the set I6N , for i ¼ 4; . . . ; 13.
This action is well defined since Gi is a subgroup of the point
group of the lattice, and therefore lattice points are mapped
into lattice points under elements of Gi. Let DðiÞN ¼
fvðiÞ1 ; . . . ; vðiÞkig be a set of distinct representatives for the orbits
of Gi in I6N . Since G4 & Gi for all i ¼ 5; . . . ; 13, and thus their
fundamental domains are contained in that of G4, the set Dð4ÞN
contains the sets of representatives DðiÞN for the groups Gi,
i ¼ 5; . . . ; 13, which are not necessarily distinct. Since we do
not have information on the group G4 apart from its genera-
tors, the set Dð4ÞN is computed numerically according to the
following procedure:
(i) For v 2 I6N , compute OG4ðvÞ.
(ii) Among all vi 2 OG4ðvÞ identify vv with the largest
number of positive components, choosing at random if two or
more points fulfil this property.
(iii) Add vv to Dð4ÞN and repeat from the start until all v 2 I6N
have been considered.
In particular, Dð4ÞN thus obtained contains 47, 183 and 529
points for N ¼ 2; 3 and 4, respectively. With this setup, the
library is given by
SðNÞ :¼ f#kðOGjðvÞÞg : v 2 D
ð4ÞN ; j ¼ 4; . . . ; 13
n o
; ð26Þ
which by construction consists of distinct point arrays.
Once the set SðNÞ is computed for a chosen value of N, we
retrieve the information of the viral capsid in consideration
from the VIPER data bank (Carrillo-Tripp et al., 2009). These
PDB files contain structural data of the viral capsid, such as
the coordinates of the atomic positions of the capsid proteins
and in many cases also of the packaged genome. Following
Wardman (2012), we represent the atomic positions of the
proteins by spheres of radius 1.9 A in the visualization tool
PyMOL. In order to compare the point arrays with biological
data, and hence find those point sets which best represent the
capsid features, we use the following procedure:
(i) For any group Gi 2 AII , we compute with GAP a
transversal TðiÞ ¼ ðgðiÞ1 ; . . . ; g
ðiÞniÞ for the right cosets of II in Gi,
where ni denotes the index of I in Gi.
(ii) Given a point array #kðOGiðvÞÞ 2 SðNÞ, we compute the
set
LðiÞðvÞ ¼ fj#kðgðiÞj vÞj : j ¼ 1; . . . ; nig:
The cardinality of LðiÞðvÞ is the number of distinct radial levels
in the point set #kðOGiðvÞÞ. We denote by RðiÞ
maxðvÞ :¼
maxjLðiÞðvÞ the largest radial level which corresponds to the
outermost layer in the nesting. This is used to scale the point
set so that the capsid is contained in the convex hull of the
projected orbit.
(iii) The rescaled orbit is then compared with the data in the
PDB file. We start by selecting those point arrays whose
outermost layer best represents the outermost features of the
capsid. Specifically, we consider a coarse-grained representa-
tion of the capsid surface by locating the most radially
distal clusters of C' atoms using the procedure described
by Wardman (2012). Denoting these clusters by Ck;
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Figure 3Projected orbit of the lattice point v ¼ ð0; 0; 1; 1; 2; 1Þ under the groupG4. Each layer in the resulting nested point set possesses achiralicosahedral symmetry.
k ¼ 1; . . . ;M, the Ck can be approximated byM spheres Bkð~rrÞ
of radius ~rr (for the numerical implementation, we chose the
cut-off ~rr ¼10 A). For any orbit #kðOGiðvÞÞ, we isolate its
external point layer LðoutÞ by computing the points situated at
distance Rmax (introduced above) from the origin. The orbit is
then selected if, for every point x 2 LðoutÞ, there exists
k 2 f1; . . . ;Mg such that x 2 Bkð~rrÞ.
(iv) Among the point sets thus selected, we determine those
that best match the other capsid features. For this, we isolate
the inner radial levels using the decomposition of orbits into
cosets and compare them with the location of the genomic
material and the inner capsid surface. The cardinalities of the
point arrays are not large enough to match with atomic posi-
tions, but they rather map around material as in Keef et al.
(2012); this comparison can be achieved via visual inspection
using the surface representation of the capsid in PyMOL.
We consider here two case studies: Pariacoto Virus and
Bacteriophage MS2, both T ¼ 3 capsids in the Caspar–Klug
classification. These were chosen in order to facilitate
comparison with Keef et al. (2012), where point arrays derived
from affine extensions of the icosahedral group were used to
generate blueprints for viral architecture, and Janner (2011b),
where virus architecture is approximated by lattices.
6.1. Pariacoto Virus
Pariacoto Virus (PaV) is a single-stranded RNA insect
virus, whose X-ray crystal structure reveals approximately
35% of the RNA organized as a dodecahedral cage of duplex
RNA in proximity to the inner capsid surface (Tang et al.,
2001). A characteristic feature of this capsid are the 60
protrusions of approximately 15 A around the quasi-threefold
axes, each formed by three interdigitated subunits. These are
the outermost capsid features that we will match to the largest
radial levels in the point arrays of our library in order to
identify the best-fit point array. For this we performed the
procedure described above, and found that the best fit for this
capsid is given by the projected orbit of the lattice point
vv ¼ ð2; 1;%1;%1; 0; 0Þ under the group G6 (see Fig. 4). This
point set consists of 960 points, arranged into eight radial
levels. The outermost level is formed by 60 points which map
onto the spikes at the 60 local threefold axes, see Fig. 4(b). The
third radial level from the origin describes the organization of
the RNA inside the capsid. This set is made up of 120 points
forming a truncated icosidodecahedron, which maps around
the dodecahedral RNA cage, see Fig. 4(d). The fifth radial
level from the origin, located between the RNA and the
spikes, consists of 120 points, organized into ten and 12 clus-
ters of six and five points each, which are located around the
three- and fivefold axes, respectively. In particular, we show in
Fig. 4(c) a close-up view of the clusters with fivefold symmetry.
Note that these points provide constraints on the lengths
of the protein helices and the positions of the protein subunits
of type C.
We point out thatG6 is the group of smallest order in the set
AII that provides a blueprint for PaV that captures the loca-
tion of both capsid proteins and the RNA collectively. The
orbit of vv under G4 in projection, which by construction is
contained in #kðOG6ðvvÞÞ, maps around the spikes, but totally
lacks information on the organization of the genomic material
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Figure 4Blueprints for the capsid of Pariacoto Virus (based on PDB file 1f8v, Tang et al., 2001). (a) Cross section of the capsid superimposed with the projectedorbit of vv ¼ ð2; 1;%1;%1; 0; 0Þ under the group G6. The point set consists of 960 points situated at eight distinct radial levels which provides constraintson the capsid architecture. (b) Close-up view of the outermost layer of the projected orbit, which encodes the locations of the spikes around the quasi-threefold axes. (c) The layers between the spikes and the genomic material map around the inner surface of the capsid proteins. (d) The third farthestlayer from the origin gives information on RNA organization: the 120 points, forming a truncated icosidodecahedron, map around the dodecahedralRNA cage.
inside. Moreover, all the orbits of vv under the G6-containing
Gk 2 AII , i.e. G8 and G12, as well as B6 (cf. Fig. 2) coincide in
projection, implying that they contain no additional informa-
tion on capsid architecture. Hence G6 can be chosen as the
six-dimensional symmetry group that induces the three-
dimensional structure of the PaV capsid in projection.
6.2. Bacteriophage MS2
Like PaV, MS2 is a single-stranded RNAvirus, with a T ¼ 3
capsid. Cryo-electron microscopy reveals a double-shell
structure in the organization of the genomic RNA (Toropova
et al., 2008), and we will demonstrate here that our approach is
able to capture this. With our procedure as above, we found
that the projected orbit of ~vv ¼ ð2; 1; 1;%1; 0; 1Þ under the
group G4 is the point set that provides the best blueprint for
the capsid (see Fig. 5). Specifically, this orbit contains 960
points, that are arranged, in projection, into nine radial levels.
The two outermost layers, Lð9Þ and Lð8Þ, map to the exterior of
the capsid: Lð9Þ consists of 60 points, arranged into 12 clusters
of five points each, positioned around the fivefold symmetry
axes of the capsid, whereas Lð8Þ has size 120 and is made up of
20 clusters of six points, located around the threefold axes.
This is consistent with the quasi-equivalent structure of the
T ¼ 3 capsid. We point out that Lð8Þ and Lð9Þ are in fact almost
situated at the same radial level (the ratio of their radii is
’ 1:064814), and collectively map around the capsid exterior
as demonstrated in the close-up in Fig. 5(b).
All other points of the array are from a mathematical point
of view related to these outermost shells, and should therefore
also map around material boundaries. We start by comparing
the point array with the icosahedrally averaged cryo-electron
microscopy structure at 8 A resolution in Toropova et al.
(2008). As shown in Fig. 5(a), the innermost radial levels of the
point array define the interior of the
inner RNA shell. Moreover, there are
points mapping around the outer and
inner surfaces of the other shell. There
is a layer of points between the shells
that at a first glance seems to float in
space, but a close inspection of the data
set reveals that they are in fact posi-
tioned around the RNA connecting the
outer and inner shell (see also the close-
up in Fig. 5c). This icosahedrally aver-
aged data set has been obtained via a
superposition of a large number of viral
particles, aligned according to their
symmetry axes, in order to enhance the
resolution. However, in any individual
particle, the RNA is organized in an
asymmetric way, that is consistent with
the icosahedrally averaged density.
Since our point arrays are not fully
constraining the structure, but are
providing blueprints for the overall
organization of the virus, we expect the
asymmetric organization to be consistent with our symmetric
point arrays. In order to test this hypothesis, we compare our
model with the asymmetric RNA density of a cryo-electron
microscopy tomogram at about 39 A resolution (Dent et al.,
2013; Geraets et al., 2015), see Fig. 6. Since the density is shown
in a cross-sectional view, the density in the two shells cannot
be seen in full. However, as expected, the density is consistent
with the radial levels defined by the point arrays, consistent
with our hypothesis that the mathematical model indeed
describes material boundaries in this virus. Taken together,
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Acta Cryst. (2015). A71, 569–582 Reidun Twarock et al. ! Orbits of non-crystallographic groups and viruses 579
Figure 5The projected orbits of ~vv ¼ ð2; 1; 1;%1; 0; 1Þ under the group G4 provide blueprints for the capsidof Bacteriophage MS2 (based on PDB file 1aq3, van den Worm et al., 1998). (a) Cross section of thecapsid: the point set consists of nine different radial levels which encode information on the positionof capsid proteins and the genomic material of the virus. (b) Close-up view of the outermost layersof the projected orbit which map around the capsid proteins. (c) Close-up view of the RNA density.The second and third innermost layers (in blue and green, respectively) map around the fivefoldsymmetry axes and connect the two RNA shells.
Figure 6A cryo-tomogram of Bacteriophage MS2, based on an RNA density mapprovided by Dent et al. (2013), superimposed on the best-fit point arrayfor Bacteriophage MS2. The top of the figure shows a portion of thebacterial pilus, the natural receptor of this virus. The surface representa-tion shows both the capsid on the exterior and the genomic RNA. Theinner and outer RNA shells follow the blueprints of the array points, butrealize it in an asymmetric way as expected.
these results imply that the group G4 is the group of smallest
order in our classification that provides structural constraints
on the capsid proteins and the genome organization of MS2,
and is therefore the symmetry group in six dimensions that
describes the structure of this virus in projection.
7. Conclusion
The method presented here is a new way of constructing finite
nested point sets with non-crystallographic symmetry from
group theoretical principles. It complements previous studies,
in which such point sets were constructed via affine extensions
of non-crystallographic groups. The latter, being based on the
theory of infinite-dimensional Kac–Moody algebras, produced
infinite point sets, and a cut-off parameter had to be intro-
duced in order to obtain finite structures. This implies that the
point sets do not correspond to orbits of finite groups. The
method developed in this paper, on the other hand, provides
for the first time a characterization of non-crystallographic
finite multi-shell structures which is entirely based on the
theory of finite groups in a higher-dimensional space
which admits a crystallographic embedding of the non-
crystallographic symmetry.
With application to viruses in mind, we discussed the case of
icosahedral symmetry in detail and provided a classification of
all the subgroup chains of the hyperoctahedral group that
contain a crystallographic embedding of the icosahedral
group. We showed that the point sets induced by orbits of
lattice points under such groups via projection into a three-
dimensional invariant subspace provide a library of structural
constraints on the structural organization of viruses. In parti-
cular, we presented two case studies, Pariacoto Virus and
Bacteriophage MS2, both T ¼ 3 viruses, and showed that the
corresponding constraint sets indicate material boundaries in
these viruses. We note also that previous approaches provided
good approximation for the material boundaries (Janner,
2011b; Keef et al., 2012; Salthouse et al., 2015); however, in
contrast to these approaches, we approximate here virus
architecture via point arrays that are generically finite,
because they stem from orbits of finite groups. As already
pointed out, the point sets display achiral icosahedral
symmetry at each radial level, and hence they are invariant
under reflections, i.e. the non-crystallographic Coxeter group
H3. However, since the point arrays only provide constraints
on the structural organization of a virus, but do not fully
determine its structure, this does not imply that the virus must
have full H3 symmetry. Indeed, as we have discussed above,
viruses may realize the blueprints given by the point arrays in
an asymmetric way. This can occur, e.g., via asymmetric
components in the viral capsid such as the one copy of a
maturation protein that is believed to replace one of the
protein dimers in the capsid shell of Bacteriophage MS2 (Dent
et al., 2013), or by the way in which genomic material realizes
the polyhedral genome organization observed via cryo-
electron microscopy. As an example of the latter, we discussed
a cryo-electron microscopy tomogram of the packaged RNA
of Bacteriophage MS2. However, knowledge of the possible
blueprints is important, as it can be used, in combination with
other techniques, in the analysis of low-resolution data of the
genome organization in viruses (Geraets et al., 2015).
Viruses are known to exhibit icosahedral symmetry in their
capsids due to the principle of genetic economy: the use of
symmetry in the capsid organization enables viruses to code
for a small number of different types of building blocks, thus
minimizing genome length, whilst building containers with a
maximal number of repeats (corresponding to the order of the
symmetry group) of the basic building blocks, thus achieving
maximal container volume. The high level of symmetry that is
observed at different radial levels, including genome organi-
zation, may seem surprising. However, the fact that the
interaction sites between genomic RNA and capsid proteins
are at symmetry-related positions with reference to capsid
architecture may provide an explanation for the correlation
between capsid architecture and genome organization in
terms of local interactions.
Moreover, our analysis of the group theoretical under-
pinnings of viral architecture has implications for our under-
standing of the dynamic properties of viruses. For example, it
provides a framework for the analysis of conformational
changes in the viral capsid, which are structural rearrange-
ments of the capsid proteins that are important for larger
classes of viruses to become infective. Specifically, such
structural transitions can be modelled with a crystallographic
approach, using a generalization of the concept of Bain strain
for multi-dimensional lattices (Indelicato et al., 2011) or in the
framework of the Ginzburg–Landau theory of phase transi-
tions (Zappa et al., 2013). Our work opens up a new avenue for
a description of such structural transitions in terms of
Hamiltonians that are formulated in terms of the six-
dimensional symmetry groups that induce the three-
dimensional structures of the virus in projection.
More generally, previous mathematical work on affine
extensions of non-crystallographic symmetries has resulted in
applications beyond the area of virology for which these
concepts had originally been introduced. For example, the
organization of different fullerene shells of carbon onions has
been modelled with previous approaches (Janner, 2014;
Dechant et al., 2014), and we expect that our new approach
should be relevant in this context as well. Moreover, a math-
ematical formulation of systems with non-crystallographic
symmetries is a challenge in wider areas of physics, such
as integrable systems, where models in terms of non-
crystallographic root systems have been introduced (Fring &
Korff, 2005, 2006); we expect that the use of projections of
the higher-dimensional symmetry groups, that contain non-
crystallographic symmetries as crystallographic embeddings,
could provide a new perspective also in this context.
APPENDIX A
In this Appendix we provide the generators of the
subgroupsGi, for i ¼ 1; . . . ; 13, which constitute the setAII (cf.
x5). For the computations in GAP, it is convenient to work
580 Reidun Twarock et al. ! Orbits of non-crystallographic groups and viruses Acta Cryst. (2015). A71, 569–582
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with permutation instead of matrix representations. In parti-
cular, the hyperoctahedral group is isomorphic to a subgroup
of the symmetric group S12. We briefly recall this result here;
for more details, we refer to Baake (1984).
Let a 2 Z62 and let # be a permutation in the symmetric
group S6. The set fða;#Þ : a 2 Z62;# 2 S6g is a group under the
multiplication
ða;#Þðb; %Þ :¼ ða% þ2 b;#%Þ; ða%Þk :¼ a%ðkÞ; k ¼ 1; . . . ; 6;
known as the wreath product of Z2 and S6 and denoted by
Z2 o S6. It is isomorphic to B6 via the function T : Z2 o S6 ! B6
given by
½Tða;#Þ)ij ¼ ð%1Þaj)i;#ðjÞ; i; j ¼ 1; . . . ; 6:
The function ’ : Z2 o S6 ! S12 given by
’ða;#ÞðkÞ :¼#ðkÞ þ 6ak if 1 * k * 6
#ðk% 6Þ þ 6ð1% ak%6Þ if 7 * k * 12
)
is an injective homomorphism; the composition
’ 3 T%1 : B6 ! S12 can be used to map B6 into a subgroup of
S12. In particular, the generators of B6 are given by
B6 ’ hð1; 2Þð7; 8Þ; ð1; 2; 3; 4; 5; 6Þð7; 8; 9; 10; 11; 12Þ; ð6; 12Þi;
and for the representation II of I we have [cf. equation (22)]:
II ’ hð1; 6Þð2; 5Þð3; 9Þð4; 10Þð7; 12Þð8; 11Þ;ð1; 5; 6Þð2; 9; 4Þð7; 11; 12Þð3; 10; 8Þi:
With these results, Algorithm 5.1 was implemented in GAP.
The generators for the groups Gi, for i ¼ 2; . . . ; 12, are the
following:
G2 ¼ hð1; 6Þð2; 5Þð3; 9Þð4; 10Þð7; 12Þð8; 11Þ;ð1; 5; 6Þð2; 9; 4Þð7; 11; 12Þð3; 10; 8Þ;
ð1; 7Þð2; 8Þð3; 9Þð4; 10Þð5; 11Þð6; 12Þi;
G3 ¼ hð3; 11Þð4; 12Þð5; 9Þð6; 10Þ; ð2; 3; 5; 4Þð6; 12Þð8; 9; 11; 10Þ;ð1; 2Þð3; 5Þð7; 8Þð9; 11Þi;
G4 ¼ hð1; 3Þð2; 8Þð4; 5; 10; 11Þð7; 9Þ;ð1; 3; 4; 7; 9; 10Þð2; 5; 12; 8; 11; 6Þi;
G5 ¼ hð1; 8; 9; 7; 2; 3Þð4; 6; 5Þð10; 12; 11Þ;ð1; 2Þð3; 5Þð7; 8Þð9; 11Þ; ð4; 10Þi;
G6 ¼ hð3; 9Þð6; 12Þ; ð3; 4; 5; 6Þð9; 10; 11; 12Þ;ð1; 7Þð6; 12Þ; ð1; 2; 9; 10; 11; 7; 8; 3; 4; 5Þð6; 12Þi;
G7 ¼ hð1; 7Þð6; 12Þ; ð2; 8Þð6; 12Þ;ð1; 2; 9; 10; 11; 7; 8; 3; 4; 5Þð6; 12Þ;
ð3; 4; 5; 12; 9; 10; 11; 6Þi;
G8 ¼ hð1; 8; 9; 7; 2; 3Þð4; 6; 5Þð10; 12; 11Þ;ð1; 2Þð3; 5Þð7; 8Þð9; 11Þ; ð3; 4; 5; 6Þð9; 10; 11; 12Þ; ð4; 10Þi;
G9 ¼ hð2; 8Þð6; 12Þ; ð1; 7Þð2; 5; 3Þð6; 12Þð8; 11; 9Þ;ð1; 3; 7; 9Þð2; 12; 8; 6Þ; ð1; 3; 2; 7; 9; 8Þð4; 5; 12; 10; 11; 6Þi;
G10 ¼ hð1; 2; 6; 4; 3Þð7; 8; 12; 10; 9Þ; ð5; 11Þð6; 12Þ;ð1; 2; 6; 5; 3Þð7; 8; 12; 11; 9Þ; ð5; 12; 11; 6Þi;
G11 ¼ hð1; 8; 9; 7; 2; 3Þ; ð1; 7Þð2; 3; 4Þð8; 9; 10Þ;ð1; 7Þð2; 3; 5Þð8; 9; 11Þ; ð2; 6; 3; 5; 4Þð8; 12; 9; 11; 10Þ;
ð5; 11Þi;
G12 ¼ hð2; 8Þð6; 12Þ; ð1; 2; 6; 5; 3Þð7; 8; 12; 11; 9Þ;ð5; 6Þð11; 12Þ; ð1; 2; 6; 4; 3Þð7; 8; 12; 10; 9Þi:
Acknowledgements
We thank Briony Thomas, Eric C. Dykeman, Jessica Wardman
and Pierre-Philippe Dechant for useful discussions, and
Richard J. Bingham and James Geraets for helping in the
visualization of the cross sections of the capsids. We would
also like to thank Dr Neil Ranson and collaborators for
providing us with the RNA density of Bacteriophage MS2.
MV thanks the York Centre for Complex Systems Analysis
(YCCSA) for funding the summer project ‘The Art of
Complexity’, where part of the research for this work was
carried out. RT gratefully acknowledges a Royal Society
Leverhulme Trust Senior Research Fellowship (LT130088).
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